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Poisson algebras are, just like Lie algebras, particular cases ofLie-Rinehart algebras. The latter were introduced by Rinehart in his seminal1963 paper, where he also introduces the notion of an enveloping algebra andproves --- under some mild conditions --- that the enveloping algebra of aLie-Rinehart algebra satisfies a Poincar\'e-Birkhoff-Witt theorem (PBWtheorem). In the case of a Poisson algebra $({\mathcalA},\cdot,\{\cdot,\cdot\})$ over a commutative ring $R$ (with unit), Rinehart'sresult boils down to the statement that if $\mathcal A$ is \emph{smooth} (as analgebra), then gr$(U({\mathcal A}))$ and $\mathrm{Sym}_{\mathcalA}(\Omega({\mathcal A}))$ are isomorphic as graded algebras; in this formula,$U({\mathcal A})$ stands for the Poisson enveloping algebra of ${\mathcal A}$and $\Omega({\mathcal A})$ is the ${\mathcal A}$-module of K\"ahlerdifferentials of ${\mathcal A}$ (viewing ${\mathcal A}$ as an $R$-algebra). Inthis paper, we give several new constructions of the Poisson enveloping algebrain some general and in some particular contexts. Moreover, we show that for animportant class of \emph{singular} Poisson algebras, the PBW theorem stillholds. In geometrical terms, these Poisson algebras correspond to (singular)Poisson hypersurfaces of arbitrary smooth affine Poisson varieties.

Poisson enveloping algebras and the Poincar\'e-Birkhoff-Witt theorem